There's little doubt that potatoes would win a vegetable popularity contest by a wide margin. Their bland but distinctive flavor makes them a favorite with picky and adventurous eaters alike. They can be starchy, creamy, fluffy, mealy, crunchy, or waxy, depending on which kind you buy and how you cook them, but they (almost) always have a satisfying mouth feel.
What's more, potatoes are overachievers, perhaps more versatile in terms of cooking methods than any other vegetable -- maybe any other food. You can't eat them raw, but you can cook them about a zillion different ways, and they take well to all of them. Few other vegetables are appealing when baked or boiled and then served plain, but potatoes are. They're also super mashed, roasted, French fried, au gratin, or even grilled, braised, or stir-fried.
There are three categories of potato: waxy, starchy, and (sort-of) all-purpose. Waxy potatoes, also known as boiling potatoes, include new potatoes and red potatoes; these are fairly low in starch, which means that they keep their shape and retain a pleasantly dense, creamy texture when you cook them. Waxy potatoes are easily recognized by their thin, often red or yellow skin. On the other end of the starch spectrum are starchy potatoes, which have thick, rough, usually brown skin and cook up dry, fluffy, and yielding in texture. These are good baked (which is why they're also known as baking potatoes), fried, and mashed. In between waxy and starchy potatoes lie all-purpose potatoes, which work well with most cooking methods and are a nice textural compromise: not too fluffy, not too firm.
What to Look For
No matter what kind of potato you're buying, look for hard, smooth, unblemished spuds. Soft spots and sprouts are a sign of potatoes that are past their prime, while green patches (which are mildly toxic) indicate that a potato has been exposed to the sun. Find them a cool, dark, dry place to live (the refrigerator seems fine to me, though many people don't recommend that), and they'll keep for up to several weeks-months, even.
Most people have already eaten potatoes in their most common incarnations: mashed, baked, and deep-fried. You've probably also eaten them as home fries accompanying an omelet or pile of scrambled eggs. You may not, however, have eaten them cooked in eggs, as they are in the wonderful Spanish dish known as a tortilla. Unlike the Mexican flatbread with the same name, a Spanish tortilla consists of thinly sliced potatoes and onions slowly braised in lots of olive oil, which is then drained off. (Don't throw the unused oil away; it acquires a wonderfully aromatic flavor that can add an extra hint of savoriness to just about any other dish.) The now-soft potatoes and onion are then cooked in beaten eggs to produce a flat, frittata-like cake that's wonderful eaten warm or at room temperature as a main dish or, cut into thin wedges, as a tapas-style appetizer.
Another authentic, slightly less familiar way of serving potatoes comes to us by way of India; it's called aloo paratha, and it's essentially flaky whole wheat flatbread stuffed with spiced potatoes. This may sound like carbohydrate overkill, but it's truly wonderful and as worthwhile, to my mind, as any other home baking project. Which is not to say that it's difficult or time consuming: the dough comes together in seconds in the food processor, the filling requires only boiling and mashing, and the rolling and frying process is fun and easy, especially if you can enlist anyone to help you in the kitchen.